Bring a large saucepan of water to a boil. Add the broccoli and cook until bright green and tender-crisp, 45 seconds to 1 minute. Drain and immediately rise with cold water to stop the broccoli from cooking further.
Heat 2 teaspoons canola oil in a large nonstick skillet over medium-high heat. Add the wontons and cook until the bottom of the wontons are golden brown, about 5 minutes.
Pour ¼ cup water into the skillet, cover and steam the wontons for 2 minutes. Transfer the wontons to a large bowl. Wipe out the skillet with paper towel.
Return the skillet to the heat and add 1 teaspoons oil. Add the mushrooms and cook until tender, about 3 minutes. Transfer the mushrooms to the wonton bowl.
Heat the remaining 1 teaspoon oil to the skillet, then add the red bell pepper. Cook for 1 minute, then stir in the garlic and ginger. Cook for 30 seconds.
Pour in the stir fry sauce and cook for 1 minute, stirring constantly. Add the broccoli, wontons and mushrooms (and any accumulated liquid). Toss to warm the wontons.